We provide an elementary derivation of the one-dimensional quantum harmonic oscillator propagator, using a mix of approaches, such as path integrals, canonical operators, and ladder operators. This way we take the best of each world, and find the propagator with as few tears as possible.
One quick way to obtain the ground state wave function is to note that, since it does not have zeros,12 we can write it as . Plugging this into the Schrodinger equation, we find the solution for easily.
In the case of quadratic Lagrangians, is also quadratic. In general, this is not the case, but if we restrict it to the quadratic part of the new action, we obtain the semiclassical approximation.14
Since we only performed a translation in the variable x(t), the Jacobian of this transformation is 1.
An alternative method to find this result is by using the generating function for the Hermite polynomials, , substitute x = 0, and compare the powers of t.
This trick suppresses states for n bigger than, roughly, .